Approximation of Singularly Perturbed Parabolic Reaction-diffusion Equations with Nonsmooth Data

In this paper we consider the Dirichlet problem on a rectangle for singularly perturbed parabolic equations of reaction-diffusion type. The reduced (for ε = 0) equation is an ordinary differential equation with respect to the time variable; the singular perturbation parameter ε may take arbitrary values from the half-interval (0,1]. Assume that sufficiently weak conditions are imposed upon the coefficients and the right-hand side of the equation, and also the boundary function. More precisely, the data satisfy the Hölder continuity condition with a small exponent α and α/2 with respect to the space and time variables. To solve the problem, we use the known ε-uniform numerical method (i.e., a standard finite difference operator on piecewiseuniform fitted meshes over the axes x1 and x2) which was developed previously for problems with sufficiently smooth and compatible data. It is shown that the numerical solution converges ε-uniformly at the rate of O(N−ν +N−ν 0 ), ν = m α; here the values of N and N0 define the number of nodes in the space (with respect to each variable) and time meshes. We discuss also the behavior of local accuracy of the scheme in the case where the data of the boundary-value problem are smoother on a part of the domain of definition. 2000 Mathematics Subject Classification: 65N06; 65N22; 35B25.